The Mann–Kendall test associated with the Sen's slope is a very widely used non-parametric method for trend analysis. It requires serially uncorrelated time series, yet most of the atmospheric processes exhibit positive autocorrelation. Several prewhitening methods have therefore been designed to overcome the presence of lag-1 autocorrelation. These include a prewhitening, a detrending and/or a correction of the detrended slope and the original variance of the time series. The choice of which prewhitening method and temporal segmentation to apply has consequences for the statistical significance, the value of the slope and of the confidence limits. Here, the effects of various prewhitening methods are analyzed for seven time series comprising in situ aerosol measurements (scattering coefficient, absorption coefficient, number concentration and aerosol optical depth), Raman lidar water vapor mixing ratio, as well as tropopause and zero-degree temperature levels measured by radio-sounding. These time series are characterized by a broad variety of distributions, ranges and lag-1 autocorrelation values and vary in length between 10 and 60 years. A common way to work around the autocorrelation problem is to decrease it by averaging the data over longer time intervals than in the original time series. Thus, the second focus of this study evaluates the effect of time granularity on long-term trend analysis. Finally, a new algorithm involving three prewhitening methods is proposed in order to maximize the power of the test, to minimize the number of erroneous detected trends in the absence of a real trend and to ensure the best slope estimate for the considered length of the time series.

To estimate climate changes and to validate climatic models, long-term time series associated with statistically adapted trend analysis tools are necessary. The basic requirements needed to apply specific statistical tools are usually well described, but end-users often do not systematically test whether the properties of their time series fulfill these requirements. An inappropriate usage of the statistical tools may lead to misleading conclusions. It may also happen that a time series does not meet the complete criteria of any of the statistical tools. In that case, the statistical tool must be adapted or different methods with complementary strengths and weaknesses must be used.

The time series properties that can cause misuse of statistical tools for trend analysis primarily concern the statistical distribution, the autocorrelation, missing data or periods without measurements, the presence of seasonality, irregular sampling, the presence of negatives and the rules applied in the case of data below-detection limits. A large number of trend analysis tools such as the whole family of least mean square and generalized least square methods are parametric methods and, consequently, require normally distributed residues. Unfortunately, many atmospheric measurements, which strongly depart from the normal distribution, do not meet this requirement, so that non-parametric methods have to be used. Non-parametric techniques are commonly based on rank and assume continuous monotonic increasing or decreasing trends. The Mann–Kendall (MK) test associated with the Sen's slope is the most widely applied non-parametric trend analysis method in atmospheric and hydrologic research (Gilbert, 1987; Sirois, 1998). While it has no requirement for data distribution, it must be applied to serially independent and identically distributed variables. The second condition of homogeneity of distribution is not met if a seasonality is present, but it can be solved by using the seasonal Mann–Kendall test developed by Hirsch et al. (1982). The first condition of independence is not met if the data are autocorrelated, which is often the case for atmospheric variables that are controlled by autocorrelated physical or chemical processes. To correctly analyze autocorrelated and not normally distributed errors, two different strategies are usually applied.

The first strategy tends to decrease the amount of autocorrelation by aggregating time series into monthly, seasonal, or yearly bins or even into longer periods. However, coarser time granularities (e.g., due to longer averaging periods) do not ensure that autocorrelation is removed. Moreover, the aggregation implies a decrease in the information density in the time series, such as the diurnal or seasonal cycles, the variance of the data and to some extent the data distribution. The aggregation conditions (length of the time unit, making the time unit consistent with the observed seasonality, starting phase of the time series and the averaging method) may influence the trend results (de Jong and de Bruin, 2012; Maurya, 2013) in what is called the modifiable temporal unit problem (MTUP).

The second strategy focuses on the development of algorithms to reduce the impact of the autocorrelation artifacts on the statistical significance of the MK test and on the Sen's slope. Two kinds of algorithms are usually used: (i) the prewhitening of the data to remove the autocorrelation and (ii) inflation of the variance of the trend test statistic to take into account the number of independent measurements instead of the number of data points (the autocorrelation reduces the number of degrees of freedom in tests).

In this study, the effects of various prewhitening methods on the MK statistical significance and on the slope are analyzed for time series of in situ aerosol properties, aerosol optical depth, temperature levels (tropopause and zero-degree temperature levels) and remote sensing water vapor mixing ratios. This study also analyzes the effect of the time granularity on the MK statistical significance, on the strength of the slope and on the confidence limits of various atmospheric compounds for the atmospheric time series listed above. Additionally, a new methodology combining three prewhitening methods and called 3PW is proposed in order to handle correctly the autocorrelation without decreasing the power of the test while still computing the correct slope value.

The MK test for trends is a non-parametric method based on rank. The calculated

The MK test determines the validity of the null hypothesis

The adverse effect of positive autocorrelation in time series on the number
of type 1 errors was suggested by Tiao et al. (1990) and Hamed and Rao
(1998) and later simulated (Kulkarni and von Storch, 1995; Zhang and Zwiers, 2004; Blain, 2013; Wang et al., 2015a, b; Hardison et al., 2019). All these studies clearly showed that positive autocorrelation in time series significantly increases the number of type 1 errors, whereas prewhitening procedures increase the number of type 2 errors. Larger lag-1 autocorrelation (

A popular solution to get rid of the autocorrelation problem in the MK test is to aggregate the time series in order to decrease

Two kinds of statistical procedures were developed to correct the MK test for autocorrelation in the data. The variance correction approaches (Hamed
and Rao, 1998; Yue and Wang, 2004; Hamed, 2009; Blain, 2013) consider
inflating the variance of the

This section describes all the prewhitening methods known to the authors.
The advantages and disadvantages of each method are summarized in Table 1.
It has to be noted that, for all the methods proposed, the prewhitening can
be applied only if

Advantages and disadvantages of the MK test and of the various prewhitening methods.

Wang and Swail's (2001) TFPW method (TFPW-WS) restores the low number of
type 1 errors without decreasing the power of the test (Zhang and Zwiers,
2004). The factor

Finally, Wang et al. (2015a) proposed a further approach in order to
correct TFPW-Y for both the elevated variance of slope estimators and for
the decreased slope caused by the prewhitening. Practically, the variance of

As described in Sect. 2.2 and Table 1, each of the presented prewhitening
methods has a specific advantage: the low sensitivity to type 1 errors for
PW, the high-test power for TFPW-Y, and the unbiased slope estimate for
VCTFPW. TFPW-WS has both a low type 1 error and a high test power but requires more computing time due to the iteration process. Here, we propose
a new algorithm (3PW), described in Fig. 1, which combines the advantages of
each prewhitening method:

The

The MK test that defines the statistical significance is applied to the PW and TFPW-Y data. If both tests are ss or not ss, the trend is considered ss or not ss, respectively. If TFPW-Y is ss but not PW, the trend is considered a TFPW-Y false positive (due to the higher sensitivity to type 1 errors of TFPW-Y) and the trend has to be considered not ss. If PW is ss but TFPW-Y is not, then the trend is considered a PW false positive and the trend has to be considered not ss. The probability

The Sen's slope is then computed on the VCTFPW data in order to have an unbiased slope estimate.

Scheme of the new 3PW algorithm.

In order to have a broader view of the effects of the various PW methods,
several very different time series (Table 2) were used: three surface
in situ aerosol properties (absorption coefficient, scattering coefficient and number concentration) measured at Bondville (BND), a remote, rural
station in Illinois, USA; the aerosol optical depth (AOD) measured at
Payerne (PAY) on the Swiss plateau; the tropopause and the zero-degree
temperature levels measured by radio-sounding launched at PAY; and the water
vapor mixing ratio at 1015 m measured by remote sensing at PAY. The shortest
time series (AOD and water vapor mixing ratio) cover only 10 years of
measurements, while the longest time series cover 60 years. The three in situ aerosol properties are Johnson-distributed and diverge strongly from a
normal distribution. The other time series exhibit distributions that also
diverge from a normal distribution but to a lower extent, such that some of
them have residuals of a least mean square fit, which are normally
distributed. The values of some of the time series span over several orders
of magnitude and the scattering and absorption
coefficients time series contains negative values due to detection limit issues in very clean conditions. The time series of the zero-degree temperature level also
includes negative altitudes, since it is interpolated to altitudes lower
than sea level in the case of negative ground temperature at PAY (Bader et al., 2019). All the data have high

Description of the time series: time series with units, monitoring station, period, instrument type, original granularity, ranges (1 and 99 percentiles (1%ile and 99%ile)), mean, median and standard deviation (SD), lag-1 autocorrelation of the observations (

PSAP: particle soot absorption photometer; CLAP: continuous light absorption photometer; CPC: condensation particle counter, PFR: precision filter radiometer.

Trend analyses were applied to several periods. For all the data sets, the last 10-year period (e.g., 2009–2018 for the BND aerosol scattering coefficient) is considered first and then further possible multi-decadal
periods (e.g. the last 20 years, 1999–2018; 30 years, 1989–2018) up to 60 years for the radio-sounding time series. For the in situ aerosol properties, tests with 4- to 9-year periods are also computed in order to illustrate the problems of trend analysis on very short time series. The number of data points in the time series (

To assess the statistical significance, the two-tailed

As explained in the methodology section (Sect. 2), the trend results (e.g.,
the ss, the slopes and the
CLs) depend on a number of factors, the most important ones being the prewhitening method, the number of data points in
the time series and the presence of autocorrelation. The choice of the
prewhitening method clearly affects the ss, the slope and the
CLs. Analysis choices such as the time granularity, the length of the analyzed period and
the temporal segmentation to address seasonality affect

MK trend results (Fig. 2) of the aerosol number concentration, the aerosol scattering coefficient, the tropopause level and the AOD are plotted as a function of the time granularity for the MK test and for all the prewhitening methods. The discrepancy between the results computed with no temporal segmentation and for two different temporal segmentations to address seasonality (four meteorological seasons and 12 months) can be estimated from the inserted boxplots. The three aerosol properties exhibit decreasing trends, while the results of the tropopause level time series indicate a positive trend. The negative aerosol slopes are related to the decreasing aerosol load in western Europe and North America (Collaud Coen et al., 2020; Yoon et al., 2016). The increasing tropopause level trend is related to global warming (Xian and Homeyer, 2019). The results of the trends will not be further described and discussed, since this study is only focused on the methodology of the trend analysis.

Slope and confidence
limits computed without temporal segmentation as a function of the time granularity for MK and the five prewhitening methods (indicated by colors) for

The common features for all the time series considered here are the following.

The MK, TFPW-Y, TFPW-WS and PW-cor methods result in similar slopes.

As described in Wang et al. (2015a), the absolute value of the VCTFPW slopes lies between the TFPW and the PW slope values. The absolute value of the PW slopes is always smaller than the TFPW slope values.

Large time aggregations usually lead to
not ss

CLs are smaller for finer time granularities in the presence of ss

CLs of MK, PW and TFPW-Y, which remove the lag-1 autocorrelation without compensation for the mean values and the variances of the original time series, are smaller than for VCTFPW, PW-cor and TFPW-WS. PW-cor and TFPW-WS have the highest CLs.

The ss often decreases for coarser time granularities, occasionally leading to not ss trends for some of the prewhitening methods. PW, TFPW-WS and VCTFPW methods become not ss at finer time granularities than TFPW-Y and MK due to their lower number of false positives.

The slope discrepancies between prewhitening methods are larger than the discrepancies that occur when different temporal segmentations (months or meteorological seasons) are applied for a defined prewhitening method.

As predicted theoretically, the ss depends on the prewhitening method, with
higher ss for the MK and TFPW-Y methods that are related to higher type 1
errors (false positives), while PW and VCTFPW have a lower ss and a lower
test power. This is verified on the individual time series, e.g., for the
aerosol number concentration results presented in Fig. 3a. The yearly trend
was computed for all periods (from 5 to 24 years) at all considered time
granularities (1 d to 1 month for the meteorological season temporal
segmentation), leading to 40 trends. The results show the following.

The MK test ss without prewhitening has a median of 1, with the ss for the upper quartile and upper whisker also equal to 1 and thus within the 95 % confidence level, so that only 5 trends out of 40 evaluated (i.e., 12.5 %) are not ss.

The TFPW-Y ss has a median slightly lower than 1 and only three trends (7.5 %) outside the 95 % confidence level.

The TFPW-WS ss has a median of 0.996, which is lower than MK and TFPW-Y. The lower quartile for TFPW-WS is 0.89, which is outside the 95 % confidence level and indicates that 32.5 % of the trends are not ss.

The results of both PW and PW-cor are similar to the TFPW-WS, with a median ss of 0.995, a lower quartile of 0.84 %, and 32.5 % of the trends are not ss.

The VCTFPW ss has the lowest median (0.98), first quartile (0.83) and lower whisker (0.63), leading to 37.5 % of trends being not ss.

Similar results are found for all time series but with less difference amongst the methods when the trends are obviously present or absent and more differences for weak trends.

According to Monte Carlo simulations presented in the literature (e.g., Yue et al., 2002; Wang et al., 2015a; Hardison et al., 2019), TFPW-Y leads
to a high number of false positives. Since this study deals with measured
data, the rate of false positives is defined as trends that are ss with
TFPW-Y but not ss with PW, since the latter is the method with the lowest
rate of type 1 error. Figure 3b shows that the number of false positives
depends, as expected, on the strength of the slope and on

To obtain a better view of the weakness of each MK test, the percentage of false positives taking each of the prewhitening methods as a reference is reported in Table 3 for all the data sets. PW-cor has by definition the same ss as PW, so that their performances are given in the same column. PW has to
be used as the best reference for false positives because it is the
prewhitening method with the lowest sensitivity to type 1 errors (Zhang and
Zwiers, 2004; Yue et al., 2002; Blain, 2013; Wang et al., 2015a), whereas the consideration of the other prewhitening methods as references allows for the evaluation of the discrepancy in ss among the methods. For the decadal trends, MK, TFPW-Y and VCTFPW have 32 %–47 % of false positives
taking PW as a reference. This suggests that about two-thirds and half of the trends determined using TFPW-Y and VCTFPW, respectively, are false positives. TFPW-WS has less than 2 % of false positives, so that it can be considered to have equivalent performance to PW. For the trends on short periods, the lower amounts of false positive for MK and TFPW-Y are due to
the overestimation of the slopes with these tests (see Sect. 4.4), leading to trends that are more robust and enhanced ss. The unbiased estimate of the
VCTFPW slope produces similar amounts of errors for the short-term trends to those for the decadal trends. The percentage of false positives is similar if
TFPW-WS is considered the reference. If MK or TFPW-Y is taken as a reference, PW and TFPW-WS have a very low number of false positives independent of the length of the period, leading to the conclusion that few
cases remain uncertain. Note that 5 %–10 % of cases have different ss at the 95 % confidence level if MK or TFPW-Y is used as a reference, indicating that estimation of the ss using these two methods can have a slight impact on the results. Finally, all the prewhitening methods have a higher number of false positives if VCTFPW is considered the reference because the added slope at the end of the VCTFPW procedure is smaller than the initial slope and leads to less detectable trends. Note also that the percentage of false positives of PW and TFPW-WS remains low (

Percent of false positives for all data sets relative to a reference test for the MK tests and prewhitening methods for periods of at least 10 years (decadal trends) or smaller than 8 years.

The effects of the prewhitening method on the slope (Figs. 2 and 4) also
follow the theoretically deduced assumptions.

The slope estimated on the original data is always enhanced by the positive

Due to the detrending procedure, the absolute values of the TFPW-Y slope are larger than the PW slopes and similar to the MK slope values (Fig. 2), even if a tendency to have larger TFPW-Y than MK slopes is observed (Fig. 4b). The

Due to the corrected slope and variance, the absolute values of the VCTFPW slopes are much smaller than the TFPW-Y slopes but larger than the PW slopes.

Slope differences as a function of

Distribution of the confidence limit intervals of the slope for the trend in aerosol number concentration for all periods (5–24 years) and time granularities (1 d–1 month) as a function of the method for the meteorological season temporal segmentation. Box–whisker plotting as described for Fig. 3a.

These theoretical assumptions are validated in all cases with the ss trends analyzed in this study. The water vapor mixing ratio and the zero-degree level both have a very high autocorrelation (about 0.9 at 1 d time granularity). In such cases, the removal of the autocorrelation can lead to

Not ss trends and the absolute values of the VCTFPW slope are not always larger than PW slope values.

The slope difference among the methods depends directly on

The effects of the prewhitening method on
CLs (Fig. 5) are explained by their modification of the mean and the variance of the data. Removing the lag-1
autocorrelation leads to prewhitened data with a larger variance but lower mean than the original time series. The correcting factor of

Averaging is often used to decrease

TFPW-Y and TFPW-WS remove the autocorrelation computed from the detrended
data. Figure 6b and c show the difference in

Figure 7 presents the effect of the time granularity on ss of the trends for
the zero-degree temperature level for different periods (identified by colors) and various prewhitening methods (identified by symbols). MK and
PW-cor are not included since their ss values are nearly identical to the
TFPW-Y and PW ss values, respectively. As expected, TFPW-Y exhibits the
highest ss, followed by TFPW-WS, while PW and VCTFPW exhibit the lowest ss.
The ss always decreases at coarser time granularities for all prewhitening
methods until

Statistical significance of the trends as a function of the time granularity and prewhitening methods for the zero-degree level time series for 10-, 20- and 40-year periods without temporal segmentation to address seasonality. The horizontal red and black lines correspond to the threshold of 95 % and 90 % confidence levels, respectively.

When

The effect of the time granularity on the slope strongly correlates with the

The loss of ss with coarser time granularities is even more pronounced when
evaluated for each month or meteorological season (Fig. 8). This is due to
the lower

VCTFPW slope (dots) and CL (vertical lines) as a function of the time granularity for the division of the time series into

Figure 8 clearly shows that the coarsest time granularities enhance the
variability for the different temporal segmentation choices. For example,
the interval between the minimum and maximum slopes is 2.3 times larger for the monthly average than for the daily average for the scattering coefficient
temporally segmented into 12 months (Fig. 8a) and 3.7 times larger for the
absorption coefficient with meteorological seasons (Fig. 8b), respectively.
In some cases, the sign of the slope changes with the time granularity when
the trends are not ss. As already observed in Fig. 2, the CLs also increase with time granularity due to the decrease in

The division of the year into temporal segments is a necessary condition of
the MK test if the data exhibit a clear seasonality. Statistically, it is important to have equivalent segments with similar lengths to obtain similar

The effects of the chosen temporal segmentation to address seasonality are
presented here for the VCTFPW slope and
CLs, but they are similar for the other methods as well. The effect of including temporal segmentation on the
ss of the yearly trend is rather small, with a difference of only 2 %–3 % in the number of ss trends (not shown). The division into four meteorological seasons always results in the largest number of ss trends, while the division into 12 months is less powerful for short periods due to the low number of points for each month (

Figure 9 presents the CL intervals normalized by the trend slope as a
function of the time granularity for the aerosol scattering coefficient
without temporal segmentation (blue) or divided into monthly (green) or
meteorological seasons (red) for several periods between 5 and 24 years. Due
to the decrease in

Confidence limits of VCTFPW as a function of the time granularity for various periods of the aerosol scattering coefficient time series. Blue represents no consideration of seasonalities; red represents time segmentation into four meteorological seasons and green into 12 months. The color shading corresponds to the length of the period from 5 years (lightest) to 24 years (darkest).

In the case of a seasonal MK test, yearly trend results can be considered only if the trends are homogeneous among the temporal segments (see Sect. 2.1). The division of the time series into four meteorological seasons leads to more homogeneous trends (3 times and 25 times for decadal and short periods, respectively) at the 90 % confidence level than the division into 12 months (Table 4). Thus, if meteorological seasons correspond to the observed temporal cycle of the studied time series, then those seasons should be the preferred temporal division to consider rather than monthly divisions. Monthly segmentation could be considered when the observed variability of time series is shorter or longer than the 3-month length of a meteorological season.

Percentage of yearly trends with homogeneous temporal segments as a function of the type of segment (month or season), of the prewhitening method and of the length of the periods based on all seven time series considered in this study.

As already stipulated under Sect. 2.1, a special statistic that deviates
from the normal statistic has to be applied to compute the statistical
significance for

Figure 10 shows the effect of the reduction of the period length on the slope, the
CL and the ss for the aerosol absorption coefficient data set. The first obvious effect is that the absolute values of the slope are larger for
shorter periods and that there are large differences for both the individual months and meteorological seasons. Further, these large slopes for short
time periods are associated with high
CLs and low ss. They are due to the cumulative effects of the predominant importance of the first and last years
for short periods and to the low

VCTFPW slopes (dots) and CLs (vertical lines) as a function of various periods ending in 2018 for the daily aerosol absorption coefficient for the division of the time series into

The number of data points

Figure 11 also clearly shows that small

The effects of the temporal segmentation to address seasonality and the time
granularity on the confidence limits are primarily caused by the modification of

The main effects of the various prewhitening methods on

The CL intervals depend primarily on the number of data points and, thus, the length of the time series, choice of time granularity, and temporal segmentation to address seasonality.

The ss depends mostly on the robustness of the slope, on the number of data points, and on the prewhitening method.

The slope depends mostly on the prewhitening method, with PW leading to too low slopes and MK, TFPW-Y, TFPW-WS and PW-cor resulting in absolute values of the slope that are too high, considering VCTFPW to be an unbiased slope estimate.

The prewhitening methods presented here consider only the lag-1 autocorrelation. Atmospheric processes can, sometimes, be better represented
by a higher order of autoregressive models with ss partial correlations at
lags

Time series with a pronounced seasonality can also exhibit an

The slopes computed from the various prewhitening methods for the real
atmospheric data sets considered here exhibit a large spread, and only studies with simulated time series are able to provide insight into the
slope bias of the methods. Yue et al. (2002) show that TFPW-Y leads to a better estimate of the slope than PW, which systematically underestimated
the real slope. Zhang and Zwiers (2004) compared the MK, PW, and TFPW-WS methods for various slope and

The results of our study should be compared to the shortest periods (30 years) of the Zhang and Zwiers (2004) results, where they found an underestimation of the slope by PW and an overestimation by MK and TFPW-WS. Wang et al. (2015a) showed that the VCTFPW method leads to root mean square errors (RMSEs) of the slope lower than the RMSE for TFPW-Y slopes for all slopes and

All the simulation studies described above report slope per year based on
yearly aggregated time series. Their number of data points corresponds then
to the time series length. In contrast,

Based on the results presented in this study as well as the findings from
the literature referenced above, the following recommendations can be made.

A prewhitening method must be used on time series when

The seasonal MK test must be used on time series with a clear seasonal cycle. The chosen temporal segmentation to address seasonality for the MK test has to be compatible with the observed seasonality of the time series.

Finer time granularities should be used in order to maximize the number of data points and will yield smaller confidence limits and larger ss. The choice of the time granularity must also be compatible with the observed seasonality of the time series.

Periods shorter than 10 years must be handled with great caution and periods shorter than 8 years should not be used for long-term trend analysis.

When describing trend results, the sign of the slope should not be mentioned if it is not ss, because not ss trends cannot, by definition, be distinguished from zero trends. Moreover, not ss trends have a larger dependency on how the trends are computed (time granularity, period, prewhitening method, temporal segmentation to address seasonality, etc.).

In the presence of ss lag-1 autocorrelation, either 3PW (using both PW and TFPW-Y) or TFPW-WS should be used to assess statistical significance. MK, TFPW-Y alone and VCTFPW lead to a high number of false positives.

The slope should be corrected in order to take into account the effect of the prewhitening on the mean and the variance of the time series. We recommend the VCTFPW method to eliminate slope biases, at least for time series shorter than 30 years.

In the presence of ss trends, the confidence limits must also be considered in order to assess the uncertainty in the slope.

Several prewhitening methods, including solely prewhitening, the trend-free prewhitening from Yue et al. (2002) and from Wang and Swail (2001), as well as the variance-corrected trend-free prewhitening method of Wang et al. (2015a), were tested on seven time series of various in situ and remote sensing atmospheric measurements. Consistent with the literature, the use of MK, TFPW-Y and VCTFPW results in a large amount of false positive results, while TFPW-WS results in less than 2 % of false positives if PW is considered the reference. The power of the test is good for all the applied MK tests for the time series considered here.

The effect of choosing time granularities ranging from 1 d to 1 year was also evaluated since a common way to overcome the autocorrelation problem is to average time series to a coarser time granularity. It was found that the

Since all the time series studied exhibited clear seasonal cycles, two temporal segmentations (12 months and four meteorological seasons) were tested for the seasonal MK test. The segmentation into four meteorological seasons resulted in more homogeneous trends among the segments, a necessary condition to compute yearly trends. The division into meteorological seasons also resulted in a higher number of data points available in each temporal segment relative to division into monthly segments. No systematic effect of the choice of temporal segment on the slope was observed, and the difference between temporal segment choices was always much lower than the differences among the prewhitening methods.

Finally, a new 3PW algorithm was proposed combining several prewhitening methods to obtain a better estimate of trend and statistical significance than would be achieved with any individual prewhitening method. PW and TFPW-Y were used to compute the statistical significance of the trend and VCTFPW was applied to estimate the slope. This approach takes advantage of the low sensitivity of type 1 errors of PW, the high test power of TFPW-Y, and the less biased slope estimated by VCTFPW.

We provide, in dedicated GitHub repositories hosted within the “mannkendall” organization (

EA, GM, GR and LV performed the measurements, QC and database transfer of the time series. MCC developed the new 3PW algorithm, wrote the Matlab routines, computed the long-term trends and wrote the manuscript. FPAV and AB translated the Matlab code into Python and R, respectively. All the co-authors revised the manuscript.

The author declares that there is no conflict of interest.

The authors would like to thank Patrick Sheridan (NOAA) for mentoring and providing the Bondville data, Derek Hageman (University of Colorado) for programming efforts in data acquisition and archiving, and the on-site technical staff from the Illinois State Water Survey for their long-term support and care for the instrumentation.

This paper was edited by Mark Weber and reviewed by Wenpeng Wang and one anonymous referee.